Wavelet-based upscaling of advection equations

A method for upscaling the transport equation for flow in porous media is presented. This is a new application of the wavelet-based renormalization method for absolute permeability in Darcy’s elliptic equation for flow in porous media, described in Pancaldi et al. [V. Pancaldi, K. Christensen, P.R. King, Transp. Porous Media 67 (3) (2007) 395]. This formalism can be applied to any parabolic equation, such as the heat equation or other advection and diffusion transport equations. We present the method for a tracer transport problem. The coarse graining method consists of a rule to upscale the velocity field which determines the time-evolution of the saturation profile during immiscible displacement in two-phase flow. The technique is applied to one- and two-dimensional systems with a stochastic permeability distribution. The mean-field approximation applied neglects fluctuations in the velocity field to concentrate on the large scale behaviour of the system. Notwithstanding the restricting assumptions, this approximation provides a statistically good estimate for the motion of the saturation fronts on an upscaled grid, given the permeability map on the original fine grid. Results on one-dimensional systems are compared with analytical solutions and results for system ensembles and two-dimensional systems are presented